Simplify Quotient: Numerator, Denominator, Undefined Values

Nov 3, 2025 by Andrew McMorgan 60 views

Simplifying the Quotient: A Step-by-Step Guide

Hey guys! Let's dive into simplifying this complex quotient and figure out what's going on. We'll break it down step by step, making sure we understand each part. Our main goal here is to find the simplest form of the quotient, identify the numerator and denominator, and pinpoint those pesky values of x that make the expression undefined. So, buckle up, and let’s get started!

Breaking Down the Quotient

Simplifying quotients might seem intimidating, but it’s really about carefully factoring and canceling out common terms. The given quotient is:

(3x³ - 27x) / (2x² + 13x - 7) ÷ (3x) / (4x² - 1)

To make things easier, remember that dividing by a fraction is the same as multiplying by its reciprocal. This means we can rewrite the expression as:

(3x³ - 27x) / (2x² + 13x - 7) * (4x² - 1) / (3x)

Now, we're dealing with multiplication, which is often simpler to manage. Let’s move on to factoring each part of the expression. This is where we’ll find common terms that we can cancel out later, so pay close attention!

Factoring the Numerator and Denominator

To simplify the quotient, we need to factor each polynomial. Factoring helps us identify common terms that can be canceled out, leading to the simplest form of the expression. Let's start with the first numerator:

Factoring 3x³ - 27x

First, we look for the greatest common factor (GCF). In this case, the GCF is 3x. Factoring out 3x, we get:

3x(x² - 9)

Notice that (x² - 9) is a difference of squares, which can be further factored as:

3x(x - 3)(x + 3)

So, the factored form of the first numerator is 3x(x - 3)(x + 3). This factored form is crucial because it reveals the expression's components, making simplification easier.

Factoring 2x² + 13x - 7

Next, we factor the first denominator, 2x² + 13x - 7. This is a quadratic expression. We look for two numbers that multiply to (2 * -7 = -14) and add up to 13. Those numbers are 14 and -1. We can rewrite the middle term using these numbers:

2x² + 14x - x - 7

Now, we factor by grouping:

2x(x + 7) - 1(x + 7)

(2x - 1)(x + 7)

So, the factored form of the first denominator is (2x - 1)(x + 7). Factoring quadratic expressions like this is a fundamental skill in simplifying rational expressions.

Factoring 4x² - 1

Now let’s tackle the second numerator, 4x² - 1. This is another difference of squares, which factors as:

(2x - 1)(2x + 1)

Factoring 3x

The second denominator, 3x, is already in its simplest form. No factoring needed here!

Rewriting and Simplifying the Expression

Now that we've factored every part of the expression, let’s rewrite the entire quotient using these factored forms:

[3x(x - 3)(x + 3)] / [(2x - 1)(x + 7)] * [(2x - 1)(2x + 1)] / [3x]

This might look a bit complex, but it’s set up perfectly for simplification. We can now cancel out common factors that appear in both the numerator and the denominator. This is where the magic happens, making the expression much cleaner.

Canceling Common Factors

Look for factors that appear in both the numerator and the denominator. We can cancel out the following:

3x in the first numerator and the second denominator.
(2x - 1) in the first denominator and the second numerator.

After canceling these common factors, our expression simplifies to:

[(x - 3)(x + 3)] / [(x + 7)] * [(2x + 1)] / [1]

Multiplying the remaining terms gives us:

[(x - 3)(x + 3)(2x + 1)] / (x + 7)

So, we've successfully simplified the quotient by factoring and canceling common terms!

Identifying the Numerator and Denominator

After simplifying, we can easily identify the numerator and the denominator. This is a crucial step in understanding the structure of the simplified expression.

The Numerator

The numerator is the expression above the fraction bar. In our simplified form, the numerator is:

(x - 3)(x + 3)(2x + 1)

This is the product of three binomials. We can leave it in this factored form, or we can expand it if needed. For most purposes, the factored form is more useful because it clearly shows the roots of the polynomial.

The Denominator

The denominator is the expression below the fraction bar. In our simplified form, the denominator is:

(x + 7)

This is a simple binomial. The denominator is crucial for determining the values of x for which the expression is undefined, which we’ll discuss in the next section.

Determining Undefined Values of x

One of the most important aspects of working with rational expressions is identifying the values of x that make the expression undefined. These are the values that cause the denominator to equal zero, which is a big no-no in mathematics (we can't divide by zero, guys!).

Setting the Denominator to Zero

To find these values, we set the denominator equal to zero and solve for x:

x + 7 = 0

Subtracting 7 from both sides, we get:

x = -7

So, when x is -7, the denominator becomes zero, and the expression is undefined. This value is critical to exclude when working with the expression.

Considering Original Denominators

It's important to remember that we need to consider the denominators from the original expression as well. This is because any value of x that makes any of the original denominators zero will also make the overall expression undefined. Our original denominators were:

2x² + 13x - 7, which factored to (2x - 1)(x + 7)
3x
4x² - 1, which factored to (2x - 1)(2x + 1)

We already found that x = -7 makes the first denominator zero. Let's find the other values:

Setting 2x - 1 = 0

Adding 1 to both sides gives:

2x = 1

Dividing by 2, we get:

x = 1/2

So, x = 1/2 also makes the expression undefined.

Setting 3x = 0

Dividing by 3, we get:

x = 0

So, x = 0 is another value that makes the expression undefined.

Setting 2x + 1 = 0

Subtracting 1 from both sides gives:

2x = -1

Dividing by 2, we get:

x = -1/2

So, x = -1/2 is also a value that makes the expression undefined.

Final Undefined Values

Therefore, the expression is undefined when x is -7, 1/2, 0, or -1/2. These values must be excluded from the domain of the expression. Identifying these values is crucial for working with rational expressions and understanding their behavior.

Conclusion

We’ve successfully simplified the given quotient, identified the numerator and denominator in its simplest form, and found the values of x for which the expression is undefined. Remember, simplifying quotients involves factoring, canceling common terms, and being mindful of values that make the denominator zero. Keep practicing, and you’ll become a pro at simplifying these expressions! You got this, guys!